L(s) = 1 | + (0.625 − 0.203i)2-s + (−2.88 + 2.09i)4-s + (−2.50 + 7.70i)5-s + (−2.40 − 3.30i)7-s + (−2.92 + 4.02i)8-s + 5.32i·10-s + (−6.70 + 8.72i)11-s + (14.0 − 4.56i)13-s + (−2.17 − 1.57i)14-s + (3.39 − 10.4i)16-s + (28.0 + 9.09i)17-s + (−7.52 + 10.3i)19-s + (−8.93 − 27.4i)20-s + (−2.41 + 6.81i)22-s − 12.6·23-s + ⋯ |
L(s) = 1 | + (0.312 − 0.101i)2-s + (−0.721 + 0.524i)4-s + (−0.500 + 1.54i)5-s + (−0.343 − 0.472i)7-s + (−0.365 + 0.503i)8-s + 0.532i·10-s + (−0.609 + 0.792i)11-s + (1.08 − 0.351i)13-s + (−0.155 − 0.112i)14-s + (0.212 − 0.653i)16-s + (1.64 + 0.535i)17-s + (−0.395 + 0.544i)19-s + (−0.446 − 1.37i)20-s + (−0.109 + 0.309i)22-s − 0.549·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.596448 + 0.816798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.596448 + 0.816798i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (6.70 - 8.72i)T \) |
good | 2 | \( 1 + (-0.625 + 0.203i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (2.50 - 7.70i)T + (-20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (2.40 + 3.30i)T + (-15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (-14.0 + 4.56i)T + (136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-28.0 - 9.09i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (7.52 - 10.3i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + 12.6T + 529T^{2} \) |
| 29 | \( 1 + (7.85 + 10.8i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-10.9 - 33.8i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-14.5 + 10.5i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-9.51 + 13.0i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 44.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-7.76 - 5.64i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-17.8 - 54.9i)T + (-2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (81.1 - 58.9i)T + (1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-79.0 - 25.6i)T + (3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 87.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-21.1 + 65.1i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (15.2 + 21.0i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (62.3 - 20.2i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (83.0 + 26.9i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 - 29.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + (11.3 + 34.9i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.04791579545277983139175971003, −12.93650972220063035551316687505, −11.99582670117906767412988483919, −10.68223807682088150919588511164, −9.965269603811140349922625328022, −8.176956366271201206189930627104, −7.37496252665546859587463869944, −5.90967553642790153734373543602, −4.00793141538962074719495023967, −3.10305436425211098473422359037,
0.75318338070977472037862914743, 3.77794273581170212153817810826, 5.10915242469911922081673995371, 5.94798779138932104077833084035, 8.111924201413762310242479916553, 8.876247133669985602471140132176, 9.842761611342033587851348526830, 11.43447924174682635828349603626, 12.57585201938246534086461899311, 13.24500941261875003549400789544